Communications in Mathematical Physics

, Volume 303, Issue 3, pp 785–824 | Cite as

Concentration of Measure for Quantum States with a Fixed Expectation Value

  • Markus P. Müller
  • David GrossEmail author
  • Jens Eisert


Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors \({|\psi\rangle}\) that have a fixed expectation value \({\langle\psi|H|\psi\rangle=E}\) with respect to H. Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere.


Quantum State Isoperimetric Inequality Reduce Density Matrix Gibbs State Pure Quantum State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alon N., Spencer J.H.: The probabilistic method. Wiley, Newyork (2000)CrossRefzbMATHGoogle Scholar
  2. 2.
    Lloyd S., Pagels H.: Complexity as thermodynamic depth. Ann. Phys. 188, 186 (1988)CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Hayden P., Leung D., Winter A.: Aspects of generic entanglement. Commun. Math. Phys. 265, 95 (2006)CrossRefzbMATHADSMathSciNetGoogle Scholar
  4. 4.
    Hayden P., Leung D.W., Shor P.W., Winter A.: Randomizing quantum states: Constructions and applications. Commun. Math. Phys. 250, 371 (2004)CrossRefzbMATHADSMathSciNetGoogle Scholar
  5. 5.
    Horodecki M., Oppenheim J., Winter A.: Quantum information can be negative. Nature 436, 673 (2005)CrossRefADSGoogle Scholar
  6. 6.
    Hastings M.B.: A counterexample to additivity of minimum output entropy. Nature Phys. 5, 255 (2009)CrossRefADSGoogle Scholar
  7. 7.
    Gross D., Flammia S.T., Eisert J.: Most quantum states are too entangled to be useful as computational resources. Phys. Rev. Lett. 102, 190501 (2009)CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Bremner M.J., Mora C., Winter A.: Are random pure states useful for quantum computation. Phys. Rev. Lett. 102, 190502 (2009)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Goldstein S., Lebowitz J.L., Tumulka R., Zanghi N.: Canonical typicality. Phys. Rev. Lett. 96, 050403 (2006)CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Popescu S., Short A.J., Winter A.: Entanglement and the foundations of statistical mechanics. Nature Phys. 2, 754 (2006)CrossRefADSGoogle Scholar
  11. 11.
    Reimann P.: Foundation of statistical mechanics under experimentally realistic conditions. Phys. Rev. Lett. 101, 190403 (2008)CrossRefADSGoogle Scholar
  12. 12.
    Gogolin C.: Einselection without pointer states. Phys. Rev. E 81, 051127 (2010)CrossRefADSGoogle Scholar
  13. 13.
    Srednicki M.: Chaos and quantum thermalization. Phys. Rev. E 50, 888 (1994)CrossRefADSGoogle Scholar
  14. 14.
    Garnerone S., de Oliveira T.R., Zanardi P.: Typicality in random matrix product states. Phys. Rev. A 81, 032336 (2010)CrossRefADSGoogle Scholar
  15. 15.
    Kollath C., Läuchli A., Altman E.: Quench dynamics and non equilibrium phase diagram of the Bose-Hubbard model. Phys. Rev. B 74, 174508 (2006)CrossRefGoogle Scholar
  16. 16.
    Rigol M., Dunjko V., Yurovsky V., Olshanii M.: Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007)CrossRefADSGoogle Scholar
  17. 17.
    Cramer M., Dawson C.M., Eisert J., Osborne T.J.: Exact relaxation in a class of non-equilibrium quantum lattice systems. Phys. Rev. Lett. 100, 030602 (2008)CrossRefADSGoogle Scholar
  18. 18.
    Linden, N., Popescu, S., Short, A.J., Winter, A.: On the speed of fluctuations around thermodynamic equilibrium. [quant-ph], 2009
  19. 19.
    Brody D.C., Hook D.W., Hughston L.P.: Quantum phase transitions without thermodynamic limits. Proc. R. Soc. A 463, 2021 (2007)CrossRefzbMATHADSMathSciNetGoogle Scholar
  20. 20.
    Bender C.M., Brody D.C., Hook D.W.: Solvable model of quantum microcanonical states. J. Phys. A 38, L607 (2005)CrossRefzbMATHADSMathSciNetGoogle Scholar
  21. 21.
    Fresch B., Moro G.J.: Typicality in ensembles of quantum states: Monte Carlo sampling versus analytical approximations. J. Phys. Chem. A 113, 14502 (2009)CrossRefGoogle Scholar
  22. 22.
    Jiang, Z., Chen, Q.: Understanding Statistical Mechanics from a Quantum Point of View. In preparationGoogle Scholar
  23. 23.
    Federer H.: Geometric measure theory. Springer-Verlag, Berlin-Heidelberg-New York (1969)zbMATHGoogle Scholar
  24. 24.
    Ledoux, M.: The concentration of measure phenomenon. Mathematical Surveys and Monographs 89, Providence, RI: Amer. Math. Soc., 2001Google Scholar
  25. 25.
    Cover T.M., Thomas J.M.: Elements of information theory, Second Edition. Wiley, New York (2006)Google Scholar
  26. 26.
    Gromov, M.: Metric structures for Riemannian and Non-Riemannian spaces. Modern Birkhäuser Classics, Basel-Boston: Birkhäuser, 2007Google Scholar
  27. 27.
    Zyckowski K., Sommers H.-J.: Induced measures in the space of mixed quantum states. J. Phys. A 34(35), 7111 (2001)CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Hall M.: Random quantum correlations and density operator distributions. Phys. Lett. A 242, 123 (1998)CrossRefzbMATHADSMathSciNetGoogle Scholar
  29. 29.
    Bhatia R.: Matrix analysis. Springer, Berlin-Heidelberg-New York (1997)CrossRefGoogle Scholar
  30. 30.
    Santaló L.A.: Integral geometry and geometric probability. Addison-Wesley, Reading, MA (1972)Google Scholar
  31. 31.
    Tasaki H.: Geometry of reflective submanifolds in Riemannian symmetric spaces. J. Math. Soc. Japan 58(1), 275–297 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Schneider R., Weil W.: Stochastic and integral geometry. Springer, Reading, MA (2008)CrossRefzbMATHGoogle Scholar
  33. 33.
    Funano K.: Concentration of 1-Lipschitz Maps into an infinite dimensional p-ball with the q-distance function. Proc. Amer. Math. Soc. 137, 2407 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Funano K.: Observable concentration of mm-spaces into nonpositively curved manifolds. Geometriae Dedicata 127, 49 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Elstrodt J.: Maß–und Integrationstheorie. Springer, Reading, MA (1996)zbMATHGoogle Scholar
  36. 36.
    Milman, V.D., Schechtman, G.: Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Mathematics 1200. Reading, MA: Springer, 2001Google Scholar
  37. 37.
    Blumenson L.E.: A derivation of n-dimensional spherical coordinates.. American Mathematical Monthly 67(1), 63 (1960)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Bengtsson I., Zyczkowski K.: Geometry of quantum states - an introduction to quantum entanglement. Cambridge University Press, Cambridge (2006)CrossRefzbMATHGoogle Scholar
  39. 39.
    Dempster A.P., Kleyle R.M.: Distributions determined by cutting a simplex with hyperplanes. Ann. Math. Stat. 39(5), 1473 (1968)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Barvinok: Measure concentration in optimization. Springer, Reading, MA (2007)Google Scholar
  41. 41.
    Furuta T.: Short proof that the arithmetic mean is greater than the harmonic mean and its reverse inequality. Math Ineq and Appl. 8(4), 751 (2005)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Müller M.E.: A note on a method for generating points uniformly on N-dimensional spheres. Comm. Assoc. Comput. Mach. 2, 19 (1959)Google Scholar
  43. 43.
    Marsaglia G.: Choosing a point from the surface of a sphere. The Annals of Mathematical Statistics 43(2), 645 (1972)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Markus P. Müller
    • 1
    • 2
  • David Gross
    • 3
    • 4
    Email author
  • Jens Eisert
    • 2
    • 5
  1. 1.Institute of MathematicsTechnical University of BerlinBerlinGermany
  2. 2.Institute of Physics and AstronomyUniversity of PotsdamPotsdamGermany
  3. 3.Institute for Theoretical PhysicsLeibniz University HannoverHannoverGermany
  4. 4.Institute for Theoretical PhysicsETH ZürichZürichSwitzerland
  5. 5.Institute for Advanced Study BerlinBerlinGermany

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